When two vectors $a$ and $b$ are collinear, then $a = xb$, (where $x$ is a scalar) so in linear algebra, collinearity is a narrowly and clearly defined (and binary) concept. Two vectors -- in my understanding -- are either collinear or they are not; there are no different degrees of collinearity. If $a$ and $b$ represent distinct time series or samples in econometrics, I would therefore expect to never actually find (multi)collinearity in any empirical context, because it is close to impossible that two distinct time series or samples are exact multiples of one another. We can find correlation, and I could also imagine finding variables which are -- at least approximately -- coplanar in certain contexts, but never true collinearity.
Why then is the term multicollinearity used in the context of econometrics, when what it really seems to mean is a (statistically significant) correlation between two or more explanatory variables in a multiple regression model? Is what's meant by problems of multicollinearity that the null hypothesis of no collinearity is rejected at a certain significance level, even when we know for certain that there is strictly speaking no collinearity among the explanatory variables simply by inspecting the data? Why is multiple correlation not the more accurate term to use?
I have recently encountered the terms perfect and imperfect multicollinearity and this has confused me additionally. Does someone have a rigorous understanding of this and could share it? I would very much appreciate it!